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integral inequality; bipartite graph; graph homomorphism; Sidorenko's conjecture
We study lower estimates for integral fuctionals for which the structure of the integrand is defined by a graph, in particular, by a bipartite graph. Functionals of such kind appear in statistical mechanics and quantum chemistry in the context of Mayer's transformation and Mayer's cluster integrals. Integral functionals generated by graphs play an important role in the theory of graph limits. Specific kind of functionals generated by bipartite graphs are at the center of the famous and much studied Sidorenko's conjecture, where a certain lower bound is conjectured to hold for every bipartite graph. In the present paper we work with functionals more general and lower bounds significantly sharper than those in Sidorenko's conjecture. In his 1991 seminal paper, Sidorenko proved such sharper bounds for several classes of bipartite graphs. To obtain his result he used a certain way of ``gluing'' graphs. We prove his inequality for a new class of bipartite graphs by defining a different type of gluing.
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